Question

Question #b8258

Answer 1

The question format is not clear.
Quick answer: log of a negative is not permitted so it is a no no to `x+4<0`

Explanation:

NOT MEANT TO BE: `(fx)=log_2(x+4)-3`
This would be very different!
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`log2` would normally be understood as `log_10(2)` which is a constant.

So we have: `" "f(x)=log_10(2)(x+4)-3`

Set: `f(x)=y=log(2)x+4log(2)-3`

This is the equation of a strait line

So for this condition we have:

domain (input) `->(-oo,+oo)`
range (output) `->(-oo,+oo)`

However; as there is no 'excluded values' in this it is a little disconcerting that mention is made of an asymptote.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

So perhaps the question is meant to be: `(fx)=log_2(x+4)-3`

Set `f(x)=y=log_2(x+4)-3`

Converting this to log base 10 we have

`y=log_10(x+4)/log_10(2)-3`

Now this is different as you do have excluded values, in that, from `log_10(x+4)`

`ul("the values NOT permitted are such that")" "x+4 < 0`

So we have permitted value of

`color(red)(x>=-4" " ->" Domain"->(-4,oo)`

As `log_10(2)<1` then `log_10(x+4)/log_10(2) > log_10(x+4)`

As `x` tends to infinity then the -3 has very little effect so may be discounted

so `color(green)(lim_(x->oo)" "log_10(x+4)/log_10(2)-3=oo" "->"Part of the Range"`

When `x+4` becomes decimal then `log(x+4)` becomes negative. The magnitude of the negative number increases the closer to zero `(x-4)` becomes. Thus :

`color(green)(lim_(x+4->0)" "log_10(x+4)/log_10(2)-3= -oo" "->"Part of the range")`

`color(red)("Range "(-oo,+oo)`

Answer 2

See explanation.

Explanation:

The range is the set of all numbers for which the formula is defined.

In this example we have a `log_2` function which (as all logarythms) is defined only for positive values, so to calculate the domain we have to solve:

`x+4>0`

`x> -4`

So the domain is: `D=(-4;+oo)`

The `log_2` function takes all values, so the range is `RR`

The asymptote is `x=0`, because the closer `x` gets to zero, the smaller is `f(x)`.

`lim_{x->0} f(x)=-oo`

The graph is:

graph{ log(x+4) - 3 [-6, 30, -17.26, 5.25]}